Integral and computational representations of the

Integral Transforms and Special Functions Vol. 22, No. 7, July 2011, 487–506

Integral and computational representations of the extended Hurwitz–Lerch zeta function H.M. Srivastavaa *, Ram K. Saxenab , Tibor K. Pogányc and Ravi Saxenad a Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada; b Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342004, Rajasthan, India; c Faculty of Maritime Studies, University of Rijeka, Studentska 2, HR-51000 Rijeka, Republic of Croatia; d Faculty of Engineering, Jai Narain Vyas University, Jodhpur 342004,

Rajasthan, India (Final version received 12 June 2010 ) This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the H -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the H -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced. Keywords: Riemann zeta function; Lerch zeta function; polylogarithmic function; general Hurwitz–Lerch zeta function; gauss hypergeometric function; Fox–Wright -function; H -function; Mittag–Leffler type functions; Mellin–Barnes type integral representations; analytic continuation 2010 Mathematics Subject Classification: Primary: 11M25; 33C60; Secondary: 33C05

*Corresponding author. Email: [email protected]

ISSN 1065-2469 print/ISSN 1476-8291 online © 2011 Taylor & Francis DOI: 10.1080/10652469.2010.530128 http://www.informaworld.com

488

1.

H. M. Srivastava et al.

Introduction and preliminaries

A general Hurwitz–Lerch Zeta function (z, s, a) is defined by (see, e.g. [4, p. 27, Equation 1.11 (1)]; see also [29, p. 121 et seq.]) (z, s, a) :=

∞  n=0

zn (n + a)s

(1.1)

  a ∈ C \ Z− 0 ; s ∈ C when |z| < 1; (s) > 1 when |z| = 1 , where C is the set of complex numbers, R is the set of real numbers, R+ is the set of positive real numbers, Z is the set of integers and − Z− 0 := Z ∪ {0}

(Z− := {−1, −2, −3, . . .}).

The Hurwitz–Lerch Zeta function (z, s, a) can indeed be continued meromorphically to the whole complex s-plane, except for a simple pole at s = 1 with its residue 1. It is also known that [4, p. 27, Equation 1.11 (3)]  ∞ s−1 −at  ∞ s−1 −(a−1)t 1 t e t e 1 (z, s, a) = dt = dt (1.2) (s) 0 1 − ze−t (s) 0 et − z   (a) > 0; (s) > 0 when |z|  1 (z  = 1); (s) > 1 when z = 1 . The general Hurwitz–Lerch Zeta function (z, s, a) defined by (1.1) contains, as its special cases, not only the Riemann Zeta function ζ (s), the Hurwitz (or generalized) Zeta function ζ (s, a) and the Lerch Zeta function s (ξ ) defined by (see, for details, [4, Chapter I] and [29, Chapter 2]) ζ (s) :=

∞  n=0

ζ (s, a) :=

∞  n=0

and s (ξ ) :=

1 = (1, s, 1) (n + 1)s



1 = (1, s, a) (n + a)s



∞    e2nπiξ 2π iξ =  e , s, 1 (n + 1)s n=0

 (s) > 1 ,

(1.3)

(s) > 1; a ∈ C \ Z− 0





 (s) > 1; ξ ∈ R ,

(1.4)

(1.5)

respectively, but also other important functions of Analytic Number Theory such as the Polylogarithmic (or de Jonquère’s function) Lis (z): Lis (z) :=

∞  zn



n=1

s∈C

when

ns

= z(z, s, 1)

|z| < 1; (s) > 1

when

(1.6) 

|z| = 1

and the Lipschitz–Lerch Zeta function φ(ξ, a, s) (see [29, p. 122, Equation 2.5 (11)] and [35, p. 280, Example 8]): φ(ξ, s, a) :=  a ∈ C \ Z− 0 ; (s) > 0

∞    e2nπ iξ 2π iξ =  e , s, a (n + a)s n=0

when ξ ∈ R \ Z; (s) > 1

when

(1.7)  ξ ∈Z ,

which was first studied by Rudolf Lipschitz (1832–1903) and Matyáš Lerch (1860–1922) in connection with Dirichlet’s famous theorem on primes in arithmetic progressions.

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Yen et al. [36, p. 100, Theorem] derived the following sum-integral representation for the Hurwitz (or generalized) Zeta function ζ (s, a) defined by (1.4) 1  ζ (s, a) = (s) j =0 k−1



∞

t s−1 e−(a+j )t dt 1 − e−kt

0

(1.8)

  k ∈ N := {1, 2, 3, . . .}; (s) > 1; (a) > 0 , which, for k = 2, was given earlier by Nishimoto et al. [19, p. 94, Theorem 4]. A straightforward generalization of the sum-integral representation (1.8) was given subsequently by Lin and Srivastava [16, p. 727, Equation (7)] in the form: 1  j z (s) j =0 k−1

(z, s, a) =



∞

0

t s−1 e−(a+j )t dt 1 − zk e−kt

(1.9)

  k ∈ N; (a) > 0; (s) > 0 when |z|  1(z  = 1); (s) > 1 when z = 1 . Motivated essentially by the sum-integral representations (1.8) and (1.9), a generalization of the Hurwitz–Lerch Zeta function (z, s, a) was introduced and investigated by Lin and Srivastava [16] in the following form [16, p. 727, Equation (8)]: ) (ρ,σ μ,ν (z, s, a) :=

∞  (μ)ρn



n=0

(ν)σ n

zn (n + a)s

(1.10)

+ μ ∈ C; a, ν ∈ C \ Z− 0 ; ρ, σ ∈ R ; ρ < σ when s, z ∈ C;

 ρ = σ and s ∈ C when |z| < δ := ρ −ρ σ σ ; ρ = σ and (s − μ + ν) > 1 when |z| = δ , where (λ)ν denotes the Pochhammer symbol defined, in terms of the familiar gamma function, by  1 (ν = 0; λ ∈ C \ {0}) (λ + ν) (λ)ν := = (λ) λ(λ + 1) · · · (λ + n − 1) (ν = n ∈ N; λ ∈ C),

(1.11)

it being understood conventionally that (0)0 := 1. Clearly, we find from the definition (1.10) that ) (0,0) (σ,σ ν,ν (z, s, a) = μ,ν (z, s, a) = (z, s, a)

(1.12)

and (1,1) μ,1 (z, s, a) = ∗μ (z, s, a) :=

∞  (μ)n n=0

zn n! (n + a)s

(1.13)

  μ ∈ C; a ∈ C \ Z− 0 ; s ∈ C when |z| < 1; (s − μ) > 1 when |z| = 1 , where, as already noted by Lin and Srivastava [16], ∗μ (z, s, a) is a generalization of the Hurwitz–Lerch Zeta function considered by Goyal and Laddha [9, p. 100, Equation (1.5)]. For further results involving these classes of generalized Hurwitz–Lerch Zeta functions, see the recent works by Garg et al. [8] and Lin et al. [17].

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A generalization of the above-defined Hurwitz–Lerch Zeta functions (z, s, a) and ∗μ (z, s, a) was studied by Garg et al. [7] in the following form [7, p. 313, Equation (1.7)]: λ,μ;ν (z, s, a) := 

∞  (λ)n (μ)n n=0

zn (ν)n · n! (n + a)s

(1.14)

 λ, μ ∈ C; ν, a ∈ C \ Z− 0 ; s ∈ C when |z| < 1; (s + ν − λ − μ) > 1 when |z| = 1 .

Various integral representations and two-sided bounding inequalities for λ,μ;ν (z, s, a) can be found in the works by Garg et al. [7] and Jankov et al. [14], respectively. These latter authors [14] also considered the function λ,μ;ν (z, s, a) as a special kind of Mathieu type (a, λ)-series. By comparing the definitions (1.10) and (1.13), it is easily observed that the function λ,μ;ν (z, s, a) studied by Garg et al. [7] does not provide a generalization of the function ) (ρ,σ μ,ν (z, s, a) which was introduced earlier by Lin and Srivastava [16]. Indeed, for λ = 1, ) the function λ,μ;ν (z, s, a) coincides with a special case of the function (ρ,σ μ,ν (z, s, a) when ρ = σ = 1. μ For the Riemann–Liouville fractional derivative operator Dz defined by (see, e.g., [5, p. 181]; [21] and [15, p. 70 et seq.])  z ⎧   1 ⎪ −μ−1 ⎪ − t) f dt  < 0 (z (t) (μ) ⎪ ⎨  (−μ) 0 Dzμ {f (z)} := (1.15)  ⎪   ⎪ dm μ−m ⎪ ⎩ {f (z)} Dz m − 1   (μ) < m (m ∈ N) , dzm it is known that

  Dzμ zλ =

   (λ + 1) zλ−μ  (λ) > −1 , (1.16)  (λ − μ + 1) which, in view of the definition (1.10), yields the following fractional derivative formula for the ) generalized Hurwitz–Lerch Zeta function (ρ,σ μ,ν (z, s, a) with ρ = σ [16, p. 730, Equation (24)]:    (μ) ν−1 (σ,σ ) σ Dzμ−ν zμ−1  (zσ , s, a) = z μ,ν (z , s, a)  (ν)    (μ) > 0; σ ∈ R+ .

(1.17)

In particular, when ν = σ = 1, the fractional derivative formula (1.17) would reduce at once to the following form: ∗μ (z, s, a) =

  1 Dzμ−1 zμ−1  (z, s, a)  (μ)



 (μ) > 0 ,

(1.18)

which (as already remarked by Lin and Srivastava [16, p. 730]) exhibits the interesting (and useful) fact that ∗μ (z, s, a) is essentially a Riemann–Liouville fractional derivative of the classical Hurwitz–Lerch function  (z, s, a). Moreover, it is easily deduced from the fractional derivative formula (1.16) that  (ν) 1−λ λ−ν  λ−1 ∗ z μ (z, s, a) Dz z (λ)

  (ν) = z1−λ Dzλ−ν zλ−1 Dzμ−1 zμ−1 μ (z, s, a) , (λ)(μ)

λ,μ;ν (z, s, a) =

(1.19)

which exhibits the hitherto unnoticed fact that the function λ,μ;ν (z, s, a) studied by Garg et al. [7] is essentially a consequence of the classical Hurwitz–Lerch Zeta function (z, s, a) when we

Integral Transforms and Special Functions

491

μ

apply the Riemann–Liouville fractional derivative operator Dz two times as indicated above (see ) also many other explicit representations for ∗μ (z, s, a) and (ρ,σ μ,ν (z, s, a), which were proven by Lin and Srivastava [16], including a potentially useful Eulerian integral representation of the first kind [16, p. 731, Equation (28)]). A multiple (or, simply, n-dimentional) Hurwitz–Lerch Zeta function n (z, s, a) was studied recently by Choi et al. [3, p. 66, Equation (6)]. R˘aducanu and Srivastava (see [20] and the references cited therein), on the other hand, made use of the Hurwitz–Lerch Zeta function (z, s, a) in defining a certain linear convolution operator in their systematic investigation of various analytic function classes in Geometric Function Theory in Complex Analysis. Furthermore, Gupta et al. [11] revisited the study of the familiar Hurwitz–Lerch Zeta distribution by investigating its structural properties, reliability properties and statistical inference. These investigations by Gupta et al. [11] and others (see, e.g., [28,29,32]), fruitfully using the Hurwitz–Lerch Zeta function (z, s, a) and some of its above-mentioned generalizations, have motivated us to present here a further generalization and analogous investigation of a new family of Hurwitz–Lerch Zeta functions defined in the following form: (ρ,σ,κ)

λ,μ;ν (z, s, a) := 

∞  (λ)ρn (μ)σ n n=0

zn (ν)κn · n! (n + a)s

(1.20)

+ λ, μ ∈ C; a, ν ∈ C \ Z− 0 ; ρ, σ, κ ∈ R ; κ − ρ − σ > −1 when s, z ∈ C;

κ − ρ − σ = −1 and s ∈ C when |z| < δ ∗ := ρ −ρ σ −σ κ κ ;

 κ − ρ − σ = −1 and (s + ν − λ − μ) > 1 when |z| = δ ∗ . For the above-defined function in (1.20), we establish various integral representations, relationships with the H -function, fractional derivatives and analytic continuation formulas. In the concluding section, we also present an extension of the generalized Hurwitz–Lerch Zeta function in (1.20). (ρ,σ,κ) The following interesting special or limit cases of the function λ,μ;ν (z, s, a) defined by (1.20) are worthy of mention here. (i) For λ = ρ = 1, we find that (1,σ,κ) 1,μ;ν (z, s, a) = (σ,κ) μ,ν (z, s, a)

(1.21)

) in terms of the generalized Hurwitz–Lerch Zeta function (ρ,σ μ,ν (z, s, a) studied by Lin and Srivastava [16]. (ii) If we set ρ = σ = κ = 1, then (1.20) yields the generalized Hurwitz–Lerch Zeta function λ,μ;ν (z, s, a) studied by Garg et al. [7] and Jankov et al. [14] as follows: (1,1,1) λ,μ;ν (z, s, a) = λ,μ;ν (z, s, a).

(1.22)

(iii) Upon setting ρ = σ = κ = 1 and λ = ν, (1.20) reduces to the function ∗μ (z, s, a) studied by Goyal and Laddha [9, p. 100, Equation (1.5)] as detailed below: (1,1,1) ν,μ;ν (z, s, a) = ∗μ (z, s, a).

(1.23)

(iv) In the definition (1.20), we now set μ = ρ = σ = 1 and z  → z/λ. Then, by the familiar principle of confluence, the limit case of (1.20) when λ → ∞, would yield the Mittag–Leffler

492

H. M. Srivastava et al. (a) type function Eκ,ν (s; z) studied by Barnes (cf. [1]; see also [6, Section 18.1]), that is,

 lim

λ→∞

∞    zn 1 (1,1,κ) z (a) (s; z) λ,1;ν , s, a = =: Eκ,ν s · (ν + κn) (ν) λ (n + a) n=0   a, ν ∈ C \ Z− 0 ; (κ) > 0; s, z ∈ C ,

(1.24)

in which the parameter κ ∈ R+ has been replaced, in a rather straightforward way, by κ ∈ C with (κ) > 0. (ρ,σ,κ) (v) A limit case of the generalized Hurwitz–Lerch function λ,μ;ν (z, s, a), which is of interest in our present investigation (see Theorem 2 below), is given by ∗(σ,κ) μ;ν (z, s, a) := lim

(ρ,σ,κ)

λ,μ;ν

|λ|→∞

∞  z   (μ)σ n zn , s, a = ρ λ (ν)κn · n! (n + a)s n=0

(1.25)

 + −σ κ κ ; μ ∈ C; a, ν ∈ C \ Z− 0 ; σ, κ ∈ R ; s ∈ C when |z| < σ  (s + ν − μ) > 1 when |z| = σ −σ κ κ . (ρ,σ,κ)

(vi) Yet another limit case of the generalized Hurwitz–Lerch function λ,μ;ν (z, s, a) is given by μ∗(σ ) (z, s, a) := 



 (ρ,σ,κ)

lim

min{|λ|,|ν|}→∞

λ,μ;ν

zν κ , s, a λρ

 =

∞  (μ)σ n n=0

n!

zn (n + a)s

(1.26)

−σ μ ∈ C; a ∈ C \ Z− ; 0 ; 0 < σ < 1 and s, z ∈ C; σ = 1 and s ∈ C when |z| < σ  σ = 1 and (s − μ) > 1 when |z| = σ −σ ,

which, for σ = 1, reduces at once to the function ∗μ (z, s, a) defined by (1.13). In fact, the function μ∗(σ ) (z, s, a) defined by (1.26) can also be deduced as a special case of the (ρ,σ,κ) generalized Hurwitz–Lerch function λ,μ;ν (z, s, a) when λ = ν and ρ = κ or, alternatively, as a special case of the generalized Hurwitz–Lerch function (σ,κ) μ,ν (z, s, a) when ν = κ = 1. Thus, by comparing the series definition in (1.26) with those in (1.10) and (1.20), we get the following direct relationships: (κ,σ,κ) (σ,1) ) ∗(σ μ (z, s, a) = ν,μ;ν (z, s, a) = μ,1 (z, s, a).

(1.27)

Finally, we recall the Mittag–Leffler functions Eα (z) and Eα,β (z), which are defined by the following series: Eα (z) :=

∞  n=0

zn  (αn + 1)



z ∈ C; (α) > 0



(1.28)

and Eα,β (z) :=

∞  n=0

zn  (αn + β)



 z, β ∈ C; (α) > 0 ,

(1.29)

respectively (see also several general families of Mittag–Leffler type functions which were investigated and applied recently by Srivastava and Tomovski [31]). Clearly, therefore, we can

Integral Transforms and Special Functions

493

(a) deduce the following relationships with the Mittag–Leffler type function Eκ,ν (s; z) of Barnes [1]:  

(a) (a) (s; z) and Eα,β (z) = lim Eα,β (s; z) . (1.30) Eα (z) = lim Eα,1 s→0

s→0

More interestingly, we have

 (a) (s; z) = lim Eκ,1

κ→0

1 (z, s, a) (ν)

(1.31)

in terms of the classical Hurwitz–Lerch Zeta function (z, s, a). (a) (s; z) and the classical Mittag– Asymptotic expansions of the Mittag–Leffler type function Eκ,ν Leffler function Eα (z) defined by (1.28) are discussed by Barnes [1].

2.

Integral expressions of λ,μ;ν in terms of the Fox–Wright function p q∗ (ρ,σ,κ)

In order to present our results in this section, we need the definition of the Fox–Wright function ∗ p q (p, q ∈ N0 ) or p q (p, q ∈ N0 ), which is a generalization of the familiar generalized hypergeometric function p Fq (p, q ∈ N0 ), with p numerator parameters a1 , . . . , ap and q denominator parameters b1 , . . . , bq such that aj ∈ C

(j = 1, . . . , p) and bj ∈ C \ Z− 0

(j = 1, . . . , q),

defined by (see, for details [4, p. 183] and [30, p. 21]; see also [15, p. 56], [18, p. 30] and [33, p. 19])     ∗ (a1 , A1 ) , . . . , ap , Ap ;  z  p q (b1 , B1 ) , . . . , bq , Bq ;   ∞ (a )  1 A 1 n · · · ap A p n z n   := (b1 )B1 n · · · bq Bq n n! n=0        (b1 ) · · ·  bq (a1 , A1 ) , . . . , ap , Ap ;  z    p q = (2.1) (b1 , B1 ) , . . . , bq , Bq ;  (a1 ) · · ·  ap ⎛ ⎞ q p   ⎝Aj > 0 (j = 1, . . . , p) ; Bj > 0 (j = 1, . . . , q) ; 1 + Bj − Aj  0 ⎠ , j =1

j =1

where the equality in the convergence condition holds true for suitably bounded values of |z| given by ⎛ ⎞ ⎛ ⎞ p q   −A B |z| < ∇ := ⎝ Aj j ⎠ · ⎝ Bj j ⎠ . (2.2) j =1

j =1

In the particular case when Aj = Bk = 1

(j = 1, . . . , p; k = 1, . . . , q),

we have the following relationship (see, for details [30, p. 21]):       a , . . . , ap ; ∗ (a1 , 1) , . . . , ap , 1 ; z = p Fq 1 z p q b1 , . . . , bq ; (b1 , 1) , . . . , bq , 1 ;       (a1 , 1) , . . . , ap , 1 ;  (b1 ) · · ·  bq   p q =   z ,  (a1 ) · · ·  ap (b1 , 1) , . . . , bq , 1 ;

(2.3)

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in terms of the generalized hypergeometric function p Fq (p, q ∈ N0 ). Theorem 1 Assume that min{(a), (s)} > 0, ρ, σ, κ ∈ R+ and κ − ρ − σ  −1   with equality when |z| < δ ∗ := ρ −ρ σ −σ κ κ . Then (ρ,σ,κ) λ,μ;ν (z, s, a)

1 = (s)



∞

t

s−1 −at

e

0

∗ 2 1



 (λ, ρ), (μ, σ ); −t ze dt, (ν, κ) ;

(2.4)

provided that both sides of (2.4) exist. Proof The assertion (2.4) of Theorem 1 would follow easily from the defining series in (1.20) when we make use of the following Eulerian integral: 1 1 = (n + a)s (s)



∞

t s−1 e−(n+a)t dt



min{(s), (a)} > 0; n ∈ N0



(2.5)

0

and invert the order of summation and integration (which can be readily justified under the conditions stated with Theorem 1).  If, in Theorem 1, we set λ = ρ = 1 and apply the relationship (1.21), we obtain the following integral representation for the function (σ,κ) μ,ν (z, s, a), which was derived earlier by Lin and Srivastava [16, p. 728, Equation (14)]: (σ,κ) μ,ν (z, s, a) = 

1 (s)

 0

∞

t s−1 e−at 2 1∗



 (1, 1), (μ, σ ); −t ze dt (ν, κ) ;

(2.6)

 min{(a), (s)} > 0; κ > σ > 0 when z ∈ C; κ  σ > 0 when |z| < σ −σ κ κ ,

it being tacitly assumed that each member of (2.6) exists. For ρ = σ = κ = 1, (2.4 yields the following result due to Garg et al. [7, p. 313, Equation (2.1)]:  ∞   1 (1,1,1) λ,μ;ν (z, s, a) = λ,μ;ν (z, s, a) = t s−1 e−at 2 F1 λ, μ; ν; ze−t dt (2.7) (s) 0 which holds true under the conditions corresponding to those given with (2.4) and (2.5). Further, in view of Euler’s transformation formula [4, p. 64, Equation 2.1.4 (23)]:    a, b; c; z = (1 − z)c−a−b 2 F1 c − a, c − b; c; z   | arg(1 − z)|  π − ; 0 <  < π ; c ∈ C \ Z− 0

2 F1



(2.8)

the result (2.7) can easily be rewritten as follows:  ∞   t s−1 e−at 1 −t λ,μ;ν (z, s, a) =  λ+μ−ν 2 F1 ν − λ, ν − μ; ν; ze dt −t (s) 0 1 − ze   (a) > 0; (s) > 0 when |z|  1 (z  = 1); (s) > 1 when z = 1 ,

(2.9)

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495

which, for ν = λ, immediately yields the following known result [9, p. 100, Equation (1.6)]: 1 (s)

∗μ (z, s, a) =  (a) > 0; (s) > 0

 0

∞

t s−1 e−at μ dt 1 − ze−t



when |z|  1 (z  = 1); (s) > 1

(2.10) when

 z=1 .

This last result (2.10) was deduced also by Lin and Srivastava [16, p. 729, Equation (18)] as a special case of their integral formula (2.5). More importantly, the result (2.10) was stated by Goyal and Laddha [9, p. 100, Equation (1.6)] with the seemingly unnecessary constraint μ  1. The special case μ = 1 of the above result is precisely the well-known formula (1.2). We next present a couple of integral representation formulas for the generalized Hurwitz–Lerch (ρ,σ,κ) Zeta function λ,μ;ν (z, s, a) defined by (1.20), one of which is expressed in terms of the confluent Fox–Wright 1 1 function. Theorem 2 Let the function ∗(σ,κ) μ;ν (z, s, a) be defined by (1.25). Then (ρ,σ,κ)

λ,μ;ν (z, s, a) =

   ∞ x λ−1 zx ρ (ν) ∗(σ,κ−ρ)  , s, a dx (λ)(ν − λ) 0 (1 + x)ν μ;ν−λ (1 + x)κ   (ν) > (λ) > 0; κ  ρ > 0; σ > 0

(2.11)

and  ∞  ∞ s−1 −at λ−1 (ν) t e x (s)(λ)(ν − λ) 0 (1 + x)ν 0   (μ, σ ) ; zx ρ e−t dt dx ·1 1∗ (ν − λ, κ − ρ); (1 + x)κ   (ν) > (λ) > 0; κ  ρ > 0; σ > 0; min{(a), (s)} > 0 , (ρ,σ,κ)

λ,μ;ν (z, s, a) =

(2.12)

provided that both sides of the assertions (2.11) and (2.12) exist. Proof

Setting α = λ + ρn

and

β = ν + κn

in the Eulerian Beta-function formula: (α)(β − α) B(α, β − α) = = (β)

 0

∞

x α−1 dx (1 + x)β



 (β) > (α) > 0 ,

(2.13)

we find that  ∞ (λ)ρn x λ+ρn−1 (ν) 1 = dx (ν)κn (λ)(ν − λ) (ν − λ)(κ−ρ)n 0 (1 + x)ν+κn   (ν) > (λ) > 0; κ  ρ; n ∈ N0 ,

(2.14)

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which, by appealing to the definition (1.25), immediately yields the first assertion (2.11) of Theorem 2. Moreover, by (2.4) and (2.14), we also obtain  n  ∞ ∞  (λ)ρn (μ)σ n ze−t 1 (ρ,σ,κ) s−1 −at λ,μ;ν (z, s, a) = t e dt (s) 0 (ν)κn n! n=0  ∞  ∞ s−1 −at λ−1 t e x (ν) = ν (s)(λ)(ν − λ) 0 (1 + x) 0   ∞ n  (μ)σ n zx ρ e−t · dt dx, (ν − λ)(κ−ρ)n · n! (1 + x)κ n=0 which, in view of the definition (2.1), leads us to the second assertion (2.12) of Theorem 2.



Upon setting κ = ρ, the first assertion (2.11) of Theorem 2 immediately yields the following special case:    ∞ zx ρ x λ−1 (ν) (ρ,σ,ρ) ∗(σ )  , s, a dx (2.15) λ,μ;ν (z, s, a) = (λ)(ν − λ) 0 (1 + x)ν μ (1 + x)ρ   (ν) > (λ) > 0; σ > 0 , whose further special case when σ = 1 was given earlier by Garg et al. [7, p. 314, Equation (2.2)]. By taking κ = ρ in the second assertion (2.12) of Theorem 2, we find that  ∞  ∞ s−1 −at λ−1 t e (ν) x (ρ,σ,ρ) λ,μ;ν (z, s, a) = (s)(λ)(ν − λ) 0 (1 + x)ν 0   ρ    x (μ, σ ); −t dt dx (ν) > (λ) > 0 , (2.16) · 1 0∗ ze ; 1+x which, in the further special case when ρ = 1, yields the following result:  ∞  ∞ s−1 −at λ−1 t e (ν) x λ,μ;ν (z, s, a) = (s)(λ)(ν − λ) 0 (1 + x)ν 0 −μ  zx −t · 1− dt dx. e 1+x

(2.17)

(ρ,σ,κ)

Finally, we present a general sum-integral representation for the function λ,μ;ν (z, s, a).   Theorem 3 Let αn n∈N0 be a positive sequence such that the following infinite series: ∞ 

e−αn t

n=0

converges for any t ∈ R+ . Then (ρ,σ,κ) λ,μ;ν (z, s, a)

∞   1  ∞ s−1 −(a−α0 +αn )t  = t e 1 − e−(αn+1 −αn )t (s) n=0 0     (λ, ρ), (μ, σ ); −t ze dt min{(a), (s)} > 0 , · 2 1 (ν, κ) ;

provided that each member of the assertion (2.18) exists.

(2.18)

Integral Transforms and Special Functions

497

Proof We transform (2.4) by writing it in the form: (ρ,σ,κ) λ,μ;ν (z, s, a)

=

 (ρ,σ,κ) (ρ,σ,κ) λ,μ;ν (z, s, a − α0 + αn ) − λ,μ;ν (z, s, a − α0 + αn+1 ) .

∞   n=0

Since (by hypothesis) the infinite series: ∞ 

e−αn t

n=0 +

is convergent for any t ∈ R , the proof of the assertion (2.18) of Theorem 3 is completed by suitably applying Theorem 1. 

3.

(ρ,σ,κ)

Mellin–Barnes type integral representations for λ,μ;ν (z, s, a) and relationship of (ρ,σ,κ)

λ,μ;ν (z, s, a) with the H-function In an attempt to derive Feynman integrals in two different ways, which arise in perturbation calculations of the equilibrium properties of a magnetic mode of phase transitions, Inayat-Hussein [13, p. 4126] (see also [12]) introduced a generalization of the Fox’s H -function in the following form:    p (aj , Aj ; αj )nj=1 , (aj , Aj )j =n+1 1 m,n m,n := H (z) = H p,q [z] = H p,q z χ (s)zs ds (3.1) q 2π i (bj , Bj )m , (b , B ; β ) j j j L j =1 j =m+1 ⎛ ⎞ m n  α j " " (1 − a (b − B s) · + A s) j j j j ⎜ ⎟ √ j =1 j =1 ⎜ ⎟ , ⎜z  = 0; i = −1; χ (s) := p q β j ⎟  " " ⎝ ⎠ (aj − Aj s) · (1 − bj + Bj s) j =n+1

j =m+1

which contains fractional powers of some of the gamma functions involved. Here, and in what follows, the parameters Aj > 0

(j = 1, · · · , p)

Bj > 0 (j = 1, · · · , q),

and

the exponents αj

(j = 1, · · · , n)

and

βj

(j = m + 1, · · · , q)

can take on noninteger values and L = L(iτ ;∞) is a Mellin–Barnes type contour starting at the point τ − i∞ and terminating at the point τ + i∞ (τ ∈ R) with the usual indentations to separate one set of poles from the other set of poles. It has been established by Buschman and Srivastava [2, p. 4708] that the sufficient conditions for the absolute convergence of the contour integral in (3.1) is given by  :=

m  j =1

Bj +

n  j =1

|αj |Aj −

q 

|βj |Bj −

j =m+1

p 

Aj > 0.

(3.2)

j =n+1

The condition (3.2) provides the exponential decay of the integrand in (3.1), and the region of absolute convergence of the contour integral in (3.1) is given by | arg(z)| < where  is given by (3.2).

1 π , 2

(3.3)

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Remark 1 Functional relations for the H -function were given by Saxena [22]. A complex inversion formula for the modified H -transformation was given by Saxena and Gupta [23]. Fractional integral operators associated with the H -function were discussed by Saxena et al. (see, for details, [24,25]). Bivariate distributions associated with the H -function were considered by Saxena et al. [26]. Gupta and Soni [10] derived fractional integral formulas for the H -function. More recently, Srivastava et al. [34] presented, in a unified manner, a number of key results for the general H -function involving the Riemann–Liouville, the Weyl, and other fractional-calculus operators such as those based upon the Cauchy–Goursat Integral Formula. When α1 = · · · = αn = 1

βm+1 = · · · = βq = 1,

and

the H -function reduces to the familiar Fox’s H -function (see, for details, [18,33]). Theorem 4 The following contour integral representation holds true: (ρ,σ,κ)

λ,μ;ν (z, s, a) =

(ν) (λ)(μ)  1 (−ξ )(λ + ρξ )(μ + σ ξ ) {(ξ + a)}s · (−z)ξ dξ 2πi L (ν + κξ ) {(ξ + a + 1)}s

(3.4)

  | arg(−z)| < π , where the path of integration L is a Mellin–Barnes type contour in the complex ξ -plane, which starts at the point −i∞ and terminates at the point i∞ with indentations, if necessary, in such a manner as to separate the poles of (−ξ ) from the poles of (λ + ρξ ) and the poles of (μ + σ ξ ). Proof Let us assume that the poles of the integrand in (3.4) are simple. Calculating the residues at the poles ξ = n ∈ N0 and applying the calculus of residues, we find that ∞

(ν)  (λ + ρn)(μ + σ n) (λ)(μ) n=0 (ν + κn)



(n + a) (n + a + 1)

s

zn (ρ,σ,κ) =: λ,μ;ν (z, s, a), n! 

which evidently completes our derivation of (3.4).

Remark 2 By comparing the Mellin–Barnes contour integral in (3.4) with that in the definition (3.1), we obtain a relationship of the general Hurwitz–Lerch Zeta function (ρ,σ,κ) λ,μ;ν (z, s, a) with the H -function, which is indeed potentially useful in deriving vari(ρ,σ,κ)

ous properties of the general Hurwitz–Lerch Zeta function λ,μ;ν (z, s, a) from those of the H -function. Theorem 5 The following relationship holds true between the general Hurwitz–Lerch Zeta (ρ,σ,κ) function λ,μ;ν (z, s, a) and the H -function: (ρ,σ,κ) λ,μ;ν (z, s, a)

  (ν) 1,3 (1 − λ, ρ; 1), (1 − μ, σ ; 1), (1 − a, 1; s) −z . = H (0, 1), (1 − ν, κ; 1), (−a, 1; s) (λ)(μ) 3,3

We now consider several important and interesting special cases of Theorem 4.

(3.5)

Integral Transforms and Special Functions

499

(1) For ρ = σ = κ = 1, (3.4) reduces to the following Mellin–Barnes type integral formula:  (−ξ )(λ + ξ )(μ + ξ ) {(ξ + a)}s (ν) 1 (−z)ξ dξ, (3.6) · λ,μ;ν (z, s, a) = (λ)(μ) 2πi L (ν + ξ ) {(ξ + a + 1)}s which is valid under the conditions derivable from those associated with (3.4). In fact, the following equivalent H -function representation derivable easily as a special case of (3.5) when ρ = σ = κ = 1:   (ν) 1,3 (1 − λ, 1; 1), (1 − μ, 1; 1), (1 − a, 1; s) (1,1,1) (3.7) (z, s, a) = H 3,3 −z λ,μ;ν (0, 1), (1 − ν, 1; 1), (−a, 1; s) (λ)(μ) provides the corrected version of an obviously erroneous claim made by Garg et al. [7, p. 316, Equation (3.2)]. (2) If we set λ = ρ = 1, then (3.4) reduces to the following Mellin–Barnes type integral representation for the generalized Hurwitz–Lerch Zeta function (σ,κ) μ;ν (z, s, a) studied by Lin and Srivastava [16]:  1 (−ξ )(1 + ξ )(μ + σ ξ ) {(ξ + a)}s (ν) (σ,κ) · (−z)ξ dξ, (3.8) μ,ν (z, s, a) = (μ) 2πi L (ν + κξ ) {(ξ + a + 1)}s which is valid under the constraints derivable from those associated with (3.4). (3) For λ = ν and ρ = κ, it is found from (3.4) that  (−ξ )(μ + σ ξ ) {(ξ + a)}s 1 1 ) (z, s, a) = (−z)ξ dξ, · ∗(σ μ {(ξ + a + 1)}s (μ) 2πi L

(3.9)

) where the function ∗(σ μ (z, s, a) is defined, as before, by (1.26).

Remark 3 For σ = 1, this last result (3.9) immediately yields  1 (−ξ )(μ + ξ ) {(ξ + a)}s 1 ∗ μ (z, s, a) = · (−z)ξ dξ, {(ξ + a + 1)}s (μ) 2πi L

(3.10)

which, in its further special case when μ = 1, leads us at once to the following Mellin–Barnes type contour integral representation for the classical Hurwitz–Lerch Zeta function (z, s, a):  (−ξ )(1 + ξ ) {(ξ + a)}s 1 (−z)ξ dξ, (3.11) (z, s, a) = {(ξ + a + 1)}s 2πi L which is valid under the conditions derivable from those stated with (3.4). Remark 4 The H -function representations for the Hurwitz–Lerch Zeta function (z, s, a) and its various generalizations other than the one already covered by (3.7) can be deduced fairly easily as special or limit cases of the relationship (3.5) asserted by Theorem 5. We, therefore, choose to omit the details involved.

4.

(ρ,σ,κ)

Computational representations for λ,μ;ν (z, s, a)

In this section, we investigate an analytic continuation formula for the general Hurwitz–Lerch Zeta (ρ,σ,κ) function λ,μ;ν (z, s, a) defined by (1.20). This result will enable us to derive the corresponding

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H. M. Srivastava et al.

analytic continuation formulas for most (if not all) of the members of the families of generalized Hurwitz–Lerch Zeta and related functions. Theorem 6 The following representation holds true:     λ+n (λ + n)σ $ %n  μ− ∞   −(−z)−1/ρ (ν) ⎜ ρ ρ (ρ,σ,κ) s−1 −λ/ρ ⎜ρ   (−z) λ,μ;ν (z, s, a) = (λ + n)κ (λ)(μ) ⎝ (aρ − λ − n)s n=0 n! ν − ρ     ⎞ μ+n $ (μ + n)ρ %   λ − n ∞   −(−z)−1/σ ⎟  σ σ ⎟ | arg(−z)| < π .   + σ s−1 (−z)−μ/σ ⎠ s (μ + n)κ (aσ − μ − n) n=0 n!  ν − σ (4.1) ⎛

Proof The integral in (3.4) can easily be rewritten in the following convenient form: (ρ,σ,κ)

 1 (−ξ )(λ + ρξ )(μ + σ ξ ) (ν) (−z)ξ dξ (λ)(μ) 2πi L (ξ + a)s (ν + κξ )   | arg(−z)| < π ,

λ,μ;ν (z, s, a) =

(4.2)

where the integration contour L remains the same as the above-mentioned Mellin–Barnes type contour. We now calculate the residues of the function (λ + ρξ ) at the poles given by ξ =−

λ+n ρ

(n ∈ N0 )

and the residues of (μ + σ ξ ) at the poles given by ξ =−

μ+n σ

(n ∈ N0 ).

Thus, by the calculus of residues, we are led precisely to the asserted formula (4.1).



Corollary Under the hypotheses of Theorem 6, λ,μ;ν (z, s, a) ∼ A(−z)−λ/ρ + B(−z)−μ/σ (ρ,σ,κ)



 |z| → ∞; | arg(−z)| < π ,

(4.3)

where, for λ, μ > 0, the coefficients A and B are given by     λ μρ − λσ  ρ ρ   A= νρ − λκ s (aρ − λ) (λ)(μ) ρ



ρ s−1 (ν)

   λσ − μρ μ  σ σ .  B= νσ − μκ s (aσ − μ) (λ)(μ) σ σ s−1 (ν)

and

(4.4) We now present some corollaries and consequences of the analytic continuation formula (4.1) asserted by Theorem 6.

Integral Transforms and Special Functions

501

(a) For ρ = σ = κ = 1, (4.1) reduces to the following result for the Hurwitz–Lerch Zeta function λ,μ;ν (z, s, a) studied by Garg et al. [7]:  ∞  (λ + n)(μ − λ − n) (ν) z−n (−z)−λ (λ)(μ) n! (ν − λ − n) (a − λ − n)s n=0  ∞    (λ − μ − n)(μ + n) z−n | arg(−z)| < π , (4.5) + (−z)−μ s n! (ν − μ − n) (a − μ − n) n=0

λ,μ;ν (z, s, a) =

which holds true under the conditions derivable from those stated in Theorem 6. Furthermore, by letting s → 0 in (4.5), and using the familiar transformation formula: (λ)−n =

(−1)n (1 − λ)n

(λ ∈ C \ N; n ∈ N0 ),

we get the following well-known analytic continuation formula for the Gauss hypergeometric function [4, p. 63, Equation 2.10 (2)]   (ν)(μ − λ) 1 −λ (−z) 2 F1 λ, 1 + λ − ν; 1 + λ − μ; 2 F1 λ, μ; ν; z) = (μ)(ν − λ) z   1 (ν)(λ − μ) + (−z)−μ 2 F1 μ, 1 + μ − ν; 1 + μ − λ; (λ)(ν − μ) z   | arg(−z)| < π; |z| > 1; |z| = 1 when (ν − λ − μ) > 0 , 

(4.6)

provided that the series involved are convergent. (b) If we set ρ = λ = 1, then (4.1) reduces to the following series representation for the general Hurwitz–Lerch Zeta function studied by Lin and Srivastava [16]:    ∞   μ − (n + 1)σ z−n (ν) −1   (−z) = (μ)  ν − (n + 1)κ (a − n − 1)s n=0     μ+n μ+n $ %n   ∞  1−  −(−z)−1/σ σ σ s−1 −μ/σ   + σ (−z) (μ + n)κ (aσ − μ − n)s n=0 n!  ν − σ

(σ,κ) μ,ν (z, s, a)



 | arg(−z)| < π , (4.7)

provided that the series involved are convergent. (c) For λ = ν and ρ = κ in (4.1) or, alternatively, when we simply set ν = κ = 1 in (4.7), we obtain the following analytic continuation formula for the generalized Hurwitz–Lerch Zeta ) function ∗(σ μ (z, s, a) given by (1.26) and (1.27):  ) ∗(σ μ (z, s, a)

∞  σ s−1 (−z)−μ/σ  = (μ) n=0

μ+n σ n!



%n −(−z)−1/σ (aσ − μ − n)s $



 | arg(−z)| < π ,

(4.8) provided that the series involved are convergent. In its further special case when σ = 1, (4.8) immediately yields the following result for the generalized Hurwitz–Lerch Zeta function

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H. M. Srivastava et al.

∗μ (z, s, a) studied by Goyal and Laddha [9]: ∗μ (z, s, a) = (−z)−μ/σ

∞  (μ)n n=0



 | arg(−z)| < π ,

z−n n! (a − μ − n)s

(4.9)

provided that the series involved is convergent.

5.

(ρ,σ,κ)

Fractional derivatives of the function λ,μ;ν (z, s, a) μ

For the Riemann–Liouville fractional derivative operator Dz of the order μ, which is already defined above by (1.15), we now establish the following unification and generalization of the formulas (1.17) and (1.18). (ρ,σ,κ)

Theorem 7 The following fractional derivative formula holds true for λ,μ;ν (z, s, a):

 (ν) (ρ,σ,κ) (ρ,σ,κ) Dzν−τ zν−1 λ,μ;ν (zκ , s, a) = zτ −1 λ,μ;τ (zκ , s, a) (τ )



 (ν) > 0; κ > 0 .

(5.1)

(ρ,σ,κ)

Proof By virtue of the definition (1.20) of λ,μ;ν (z, s, a) and the familiar formula (1.16), the assertion (5.1) of Theorem 7 follows easily.  Remark 5 As we have already observed in Remark 1, Srivastava et al. [34] presented, in a unified manner, a number of key results for the general H -function involving the Riemann– Liouville, the Weyl, and other fractional-calculus operators such as those based upon the Cauchy– Goursat Integral Formula. Obviously, therefore, the assertion (5.1) of Theorem 7 can also be derived by suitably specializing one of the fractional-calculus results of Srivastava et al. [34, p. 94, Equation (2.4)], which is indeed recalled below as (6.7), by appealing appropriately to the relationship (3.5) with the H -function. The fractional derivative formula (5.1) can easily be further specialized to deduce results such as (for example) (1.17) and (1.18). For example, upon setting ρ = σ = κ = 1, (5.1) yields the following interesting formula:   (ν) τ −1 λ,μ;τ (z, s, a) z Dzν−τ zν−1 λ,μ;ν (z, s, a) = (τ )

  (ν) > 0 .

(5.2)

On the other hand, in its special case when λ = ρ = 1, we find from the assertion (5.1) of Theorem 7 that  (ν) τ −1 (σ,κ) κ  κ μ,τ (z , s, a) z Dzν−τ zν−1 (σ,κ) μ,ν (z , s, a) = (τ )



(ν) > 0; κ > 0



(5.3)

for the generalized Hurwitz–Lerch Zeta function (σ,κ) μ,ν (z, s, a) studied by Lin and Srivastava [16].

6.

(ρ,σ,κ)

Further extensions of the function λ,μ;ν (z, s, a) (ρ,σ,κ)

A natural further generalization of the function λ,μ;ν (z, s, a) can be accomplished by introducing essentially arbirary numbers of numerator and denominator parameters in the definition (1.20).

Integral Transforms and Special Functions

503

For this purpose, in addition to the symbol ∇ ∗ defined by (2.2) with, of course, Aj = ρj

(j = 1, · · · , p)

that is,

⎛ ∇ ∗ := ⎝

p 

Bj = σj

and ⎞ ⎛ −ρj

ρj

q 

⎠·⎝

j =1

(j = 1, · · · , q),

⎞ σj j ⎠ , σ

j =1

the following notations will be employed:  :=

q 

σj −

p 

j =1

and

ρj

 := s +

j =1

q 

μj −

j =1

p 

λj +

j =1

p−q . 2

(6.1)

It is fairly straightforward to present an analogous investigation of the extended Hurwitz–Lerch Zeta function (ρ ,··· ,ρ ,σ ,··· ,σ ) λ11,··· ,λpp;μ11,··· ,μqq (z, s, a) defined by p " (ρ ,··· ,ρ ,σ ,··· ,σ ) λ11,··· ,λpp;μ11,··· ,μqq (z, s, a)

=

∞  n=0



(λj )nρj

j =1 q "

n!

(μj )nσj

zn (n + a)s

(6.2)

j =1

p, q ∈ N0 ; λj ∈ C (j = 1, · · · , p); a, μj ∈ C \ Z0− (j = 1, · · · , q); ρj , σk ∈ R+ (j = 1, · · · , p; k = 1, · · · , q);  > −1 when s, z ∈ C;  = −1 and s ∈ C when

 1 ∗ |z| < ∇ ;  = −1 and () > when |z| = ∇ . 2 ∗

The special case of the definition (6.2) when p − 1 = q = 1 would obviously correspond to the (ρ,σ,κ) above-investigated generalized Hurwitz–Lerch Zeta function λ,μ;ν (z, s, a) defined by (1.20). Remark 6 If we set



p → p + 1 and q → q + 1

ρ1 = · · · = ρp = 1; λp+1 = ρp+1 = 1





 σ1 = · · · = σq = 1; μq+1 = β; σq+1 = α ,

then (6.2) reduces to the following generalized M-series which was recently introduced by Sharma and Jain [27]: α,β

∞  (a1 )k · · · (ap )k

zk (b1 )k · · · (bq )k (αk + β) k=0     (a1 , 1) , · · · , ap , 1 , (1, 1); 1 ∗   z , = p+1 q+1 (β) (b1 , 1) , · · · , bq , 1 , (β, α);

p Mq (a1 , · · · , ap ; b1 , · · · , bq ; z) =

(6.3)

in which the last relationship exhibits the fact that the so-called generalized M-series is, in fact, an obvious variant of the Fox–Wright function p q∗ defined by (2.1).

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H. M. Srivastava et al.

Each of the results involving the extended Hurwitz–Lerch Zeta function (ρ ,··· ,ρ ,σ ,··· ,σ )

λ11,··· ,λpp;μ11,··· ,μqq (z, s, a), which are asserted by Theorem 8, can be proven by applying the definition (6.2) in precisely the same manner as for the corresponding result involving the general Hurwitz–Lerch Zeta function (ρ,σ,κ) λ,μ;ν (z, s, a). Theorem 8 Each of the following relationships holds true: (ρ ,··· ,ρ ,σ ,··· ,σ )

λ11,··· ,λpp;μ11,··· ,μqq (z, s, a) =

1 (s)

 (λ1 , ρ1 ), · · · , (λp , ρp ); −t dt (6.4) ze (μ1 , σ1 ), · · · , (μq , σq ); 0   min{(a), (s)} > 0



∞

t s−1 e−at p q∗



and q " (ρ ,··· ,ρ ,σ ,··· ,σ )

λ11,··· ,λpp;μ11,··· ,μqq (z, s, a) =

j =1 p " j =1

·

1 2πi

s  (−ξ ) {(ξ + a)} L

{(ξ + a + 1)}

s



p " j =1 q "

j =1

   μj    λj

   λj + ρj ξ 

 μj + σj ξ

| arg(−z)| < π



(−z)ξ dξ

(6.5)



or, equivalently, q " (ρ ,··· ,ρ ,σ ,··· ,σ )

λ11,··· ,λpp;μ11,··· ,μqq (z, s, a) =

·

1,p+1 H p+1,q+2



j =1 p " j =1

   μj    λj

 (1 − λ1 , ρ1 ; 1), · · · , (1 − λp , ρp ; 1), (1 − a, 1; s) −z , (0, 1), (1 − μ1 , σ1 ; 1), · · · , (1 − μq , σq ; 1), (−a, 1; s)

(6.6)

provided that both sides of the assertions (6.4), (6.5) and (6.6) exist, the path of integration L in (6.5) being a Mellin–Barnes type contour in the complex ξ -plane, which starts at the point −i∞ and terminates at the point i∞ with indentations,  if necessary, in such a manner as to separate the poles of (−ξ ) from the poles of  λj + ρj ξ (j = 1, · · · , p). The H -function representation given by (6.6) can be applied in order to derive various prop(ρ ,··· ,ρ ,σ ,··· ,σ ) erties of the extended Hurwitz–Lerch Zeta function λ11,··· ,λpp;μ11,··· ,μqq (z, s, a) from those of the H -function. Thus, for example, by making use of the following fractional-calculus result due to

Integral Transforms and Special Functions

505

Srivastava et al. [34, p. 97, Equation (2.4)]:

 m,n Dzν zλ−1 H p,q (ωzκ )    n p  (1 − λ, κ; 1) , aj , Aj ; αj j =1 , aj , Aj j =n+1 m,n+1 λ−ν−1 κ  m  q =z H p+1,q+1 ωz (6.7) bj , Bj j =1 , bj , Bj ; βj j =m+1 , (1 − λ + ν, κ; 1)   (λ) > 0; κ > 0 , we readily obtain an extension of the fractional derivative formula (5.1) given by

Dzν−τ zν−1

·

 (ρ ,··· ,ρ ,σ ,··· ,σ ) λ11,··· ,λpp;μ11,··· ,μqq (zκ , s, a) =

1,p+2 H p+2,q+3

=

q " j =1 p " j =1

   μj    λj

zτ −1

 (1 − λ1 , ρ1 ; 1), · · · , (1 − λp , ρp ; 1), (1 − ν, κ; 1), (1 − a, 1; s) −z (0, 1), (1 − μ1 , σ1 ; 1), · · · , (1 − μq , σq ; 1), (1 − τ, κ; 1), (−a, 1; s)



κ

(ν) τ −1 (ρ1 ,··· ,ρp ,κ,σ1 ,··· ,σq ,κ) κ λ1 ,··· ,λp ,ν;μ1 ,··· ,μq ,τ (z , s, a) z (τ )



 (ν) > 0; κ > 0 .

(6.8)

Finally, we present the following extension of Theorem 3 above.   Theorem 9 Let αn n∈N0 be a positive sequence such that the following infinite series: ∞ 

e−αn t

n=0

converges for any t ∈ R+ . Then (ρ ,··· ,ρ ,σ ,··· ,σ ) λ11,··· ,λpp;μ11,··· ,μqq (z, s, a)

·

∗ p q



∞

1  = (s) n=0



∞

  t s−1 e−(a−α0 +αn )t 1 − e−(αn+1 −αn )t

0

 (λ1 , ρ1 ), · · · , (λp , ρp ); −t dt ze (μ1 , σ1 ), · · · , (μq , σq );



 min{(a), (s)} > 0 ,

(6.9)

provided that each member of (6.9) exists. Acknowledgements The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

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